The Quantum Mechanics Swampland
- URL: http://arxiv.org/abs/2012.11606v2
- Date: Wed, 14 Jul 2021 21:40:42 GMT
- Title: The Quantum Mechanics Swampland
- Authors: Aditya Parikh
- Abstract summary: We investigate non-relativistic quantum mechanical potentials between fermions generated by various classes of QFT operators.
We show that the potentials are nonsingular, despite the presence of terms proportional to $r-3$ and $nabla_inabla_jdelta3(vecr)$.
We propose the emphQuantum Mechanics Swampland, in which the Landscape consists of non-relativistic quantum mechanical potentials that can be UV completed to a QFT, and the Swampland consists of
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: We investigate non-relativistic quantum mechanical potentials between
fermions generated by various classes of QFT operators and evaluate their
singularity structure. These potentials can be generated either by four-fermion
operators or by the exchange of a scalar or vector mediator coupled via
renormalizable or non-renormalizable operators. In the non-relativistic regime,
solving the Schr\"odinger equation with these potentials provides an accurate
description of the scattering process. This procedure requires providing a set
of boundary conditions. We first recapitulate the procedure for setting the
boundary conditions by matching the first Born approximation in quantum
mechanics to the tree-level QFT approximation. Using this procedure, we show
that the potentials are nonsingular, despite the presence of terms proportional
to $r^{-3}$ and $\nabla_{i}\nabla_{j}\delta^{3}(\vec{r})$. This surprising
feature leads us to propose the \emph{Quantum Mechanics Swampland}, in which
the Landscape consists of non-relativistic quantum mechanical potentials that
can be UV completed to a QFT, and the Swampland consists of pathological
potentials which cannot. We identify preliminary criteria for distinguishing
potentials which reside in the Landscape from those that reside in the
Swampland. We also consider extensions to potentials in higher dimensions and
find that Coulomb potentials are nonsingular in an arbitrary number of
spacetime dimensions.
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